It's well known that 'quality' of trial/guiding wave function in DMC calculations greatly decrease variance of DMC energy and thus reduce the time required to achieve desired accuracy. Guiding wave function defines the branching term as the energy difference between true ground-state energy and the local energy of the guiding function. The better guiding wave function approximates true ground-state the less DMC energy variates.

In general, the JASTROW VMC optimized trial wave function is a suitable choice of guiding wave function in DMC. But since there are several ways to do JASTROW optimization which one gives the best result in terms of the DMC energy variance?

The quality of the trial/guiding wave function does not directly affect the final DMC estimate of the total energy of a given system (apart from the fixed node approximation). However, the intrinsic variance of VMC energy determines the variance of the estimate of the total energy at each step of the diffusion process. Therefore, as in the VMC technique, the number of DMC moves required to achieve a specific variance of the mean, decreases linearly with the intrinsic variance of VMC energy.

The quoted variance-related argument is true. However, DMC doesn't just begin and end, sampling the equilibrium distribution the whole time. That's why we have an equilibration period. If one uses an energy minimised wave function, it makes sense that this equilibration period is shorter (there are more low-energy contributions to the sum c_i \Psi_i than high energy ones). Energy minimisation maximises the amount of "c_0" in our trial wave function, and lessens the extent of the equilibration period. I would say that a very good wave function from energy minimisation (about the variance minimised starting point) is the best of both worlds - lowering both the length of the equilibration period and, in principle, the DMC variance.

It should be said that if you have highly optimised (and good!) wave functions, you might not really notice this difference very much at all. The energy and variance minima do usually occur at different points in wave function parameter space, however, if you are using unreweighted variance minimisation this difference shouldn't be massive.

As stated by J. R. Trail Heavy-tailed random error in quantum Monte Carlo VMC energy variance has Heavy-tailed distribution when using the trial wfn, which does not have an exact nodal surface. This is the usual case.VMC energy variance, moreover, does not even have an mean value, which leads to non-consistency in the process of finding the optimal JASTROW parameters by MADmin. This is especially noticeable if we investigate systems consisting of a heavy atom and one hydrogen atom like HCl or AlH, SiH radicals.

I think you are correct, however, it is nearly always the case that after an initial parameter optimisation (varmin/madmin) one would want to (before DMC) run a few cycles of Emin.

I'm not suggesting to use either variance minimisation or mad minimisation when forming your final DMC trial function.

Out of ingerest: Have you had any problems using madman in practice?

Regards, Ryan.

Ryan, since the most difficult part of JASTROW optimization is a fitting of cut-offs values in case of pure MADmin optimization I often observed how the cut-off to e-e term decreases to unphysically small value <1 au, which in turn led to an unphysical value of energy.Default limits for cut-offs are quite broad and if you narrow them, then MADmin works quite stably.

Another problem is to find such a way of JASTROW optimization, functional bases and cut-off functions that would give the minimum DMC stderr for a given number of DMC STATISTIC-ACCUMULATION steps.In this case the pure MADmin is not the best choice. I have some speculation about this, but no more.

Optimisation of the trial wave function by energy minimisation theoretically makes DMC more efficient than any other optimisation method. See David Ceperley's argument in J. Stat. Phys 43, 815 (1986).

Best wishes,

Neil.

Hello Neil. Thanks, this is a very valuable article. I can say more that VMC variance is equal to DMC uncorrelated data variance.VMC variance was taken from JASTROW optimized VMC calculations from Sample varianceDMC variance is stderr^2 * walkers * steps / N_corrEven for different ways of JASTROW optimization the dependence is the same.

On the basis of this article I will try to write down the analytical dependence for N_corr, and plot a similar graph for it. I think that in the linear approximation it is not very complicated.

UPD. For the eq. (15) form the article I've got:V = 1.71+/- 0.07*Ev/(tP) for madminV = 1.73+/- 0.03*Ev/(tP) for emin